In the sequence 8, 25, 76, 229..., each term after the first term is $b$ more than $a$ times the preceding term. What is the value of $a+b$?
Solution: From the first two terms, we have the equation \[25=8a+b.\] From the second two terms, we have the equation \[76=25a+b.\] Now we have a system of two equations in two variables, which we can solve. We isolate $b$ in the first equation to get $b=25-8a$, which we plug into the second equation to get $76=25a+25-8a$. Thus $51=17a$ and $a=3$. Now we plug this back into the first equation to solve for $b$: $25=8\cdot 3+b$ or $b=1$. Hence $a+b=3+1=\boxed{4}$.